We point out in passing that the main purpose of these tests is not to show whether one modulational approach is better than the other.
In particular, regarding the reconstruction procedure for the classical Dysthe equation,
we understand that adding contributions from higher harmonics to formulas (REF )
would likely improve their accuracy and lead to closer agreement with the full system.
Rather, a goal here is to validate our new Hamiltonian approach against other existing formulations.
As a byproduct of this comparison, given the overall positive assessment based on Figs. REF 
and REF , we in turn provide an independent validation of Curtis et al.'s model.
Such a validation was not conducted in their earlier study [1]}, [2]}.
